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# Observation of Stark many-body localization without disorder

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A Publisher Correction to this article was published on 06 January 2022

## Abstract

Thermalization is a ubiquitous process of statistical physics, in which a physical system reaches an equilibrium state that is defined by a few global properties such as temperature. Even in isolated quantum many-body systems, limited to reversible dynamics, thermalization typically prevails1. However, in these systems, there is another possibility: many-body localization (MBL) can result in preservation of a non-thermal state2,3. While disorder has long been considered an essential ingredient for this phenomenon, recent theoretical work has suggested that a quantum many-body system with a spatially increasing field—but no disorder—can also exhibit MBL4, resulting in ‘Stark MBL’5. Here we realize Stark MBL in a trapped-ion quantum simulator and demonstrate its key properties: halting of thermalization and slow propagation of correlations. Tailoring the interactions between ionic spins in an effective field gradient, we directly observe their microscopic equilibration for a variety of initial states, and we apply single-site control to measure correlations between separate regions of the spin chain. Furthermore, by engineering a varying gradient, we create a disorder-free system with coexisting long-lived thermalized and non-thermal regions. The results demonstrate the unexpected generality of MBL, with implications about the fundamental requirements for thermalization and with potential uses in engineering long-lived non-equilibrium quantum matter.

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## Data availability

The data that support the findings of this study are available from the corresponding author upon request. Source data are provided with this paper.

## Code availability

The code used for analyses is available from the corresponding author upon request.

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## Acknowledgements

We acknowledge helpful discussions with A. Migdall and R. Nandkishore. This work is supported by the DARPA Driven and Non-equilibrium Quantum Systems (DRINQS) Program (D18AC00033), the NSF Practical Fully-Connected Quantum Computer Program (PHY-1818914), the DOE Basic Energy Sciences: Materials and Chemical Sciences for Quantum Information Science program (DE-SC0019449), the DOE High Energy Physics: Quantum Information Science Enabled Discovery Program (DE-0001893), the DoE Quantum Systems Accelerator, the DOE ASCR Quantum Testbed Pathfinder program (DE-SC0019040), the DoE ASCR Accelerated Research in Quantum Computing program (DE-SC0020312), the AFOSR MURI on Dissipation Engineering in Open Quantum Systems (FA9550-19-1-0399), and the Office of Naval Research (Award N00014-20-1-2695). The authors acknowledge the University of Maryland supercomputing resources made available for conducting the research reported in this work.

## Author information

Authors

### Contributions

F.L., L.F. and W.M. proposed the experiment. W.M., P.B., K.S.C., A.K., G.P., T.Y. and C.M. contributed to experimental design, data collection and analysis. F.L. and A.V.G. contributed supporting theory and numerics. All authors contributed to the manuscript.

### Corresponding authors

Correspondence to W. Morong or F. Liu.

## Ethics declarations

### Competing interests

The authors declare competing financial interests: C.M. is co-founder and chief scientist at IonQ, Inc.

Peer review information Nature thanks the anonymous reviewers for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Experimental noise model.

a, b, Noiseless (a) and noisy (b) numerics for an initial Néel state with g/J0 = {0.24, 1.2, 1.8} (light to dark), corresponding to the data in Fig. 2c. Compared to the ideal numerics, the noisy numerics show overall lower imbalances, primarily due to the SPAM errors, and damped oscillations, primarily due to variations in the individual local effective Bz fields. However, these noise sources do not strongly affect the stability of the imbalance. c, Individual noisy realizations corresponding to the highest gradient shown above. d, Noise-averaged DEER simulations corresponding to Fig. 3b.

### Extended Data Fig. 2 Trotterization scheme.

a, Numerics comparison of the imbalance dynamics for the averaged Hamiltonian of equation (13) (solid blue line) with the full Trotter evolution (dashed orange), for the case of an initial Néel state (N = 15) and parameters corresponding to the strongest experimental field gradient. b, Difference (averaged - Trotter) between the plots in a, showing that the Trotter error over experimental timescales is on the order of one percent. c, Experimental examples (top row) of continuous and Trotterized evolution, both at g/J0 = 1.5, compared to simulations (bottom row) using the (slightly different) parameters of the individual experimental realizations. Although the Trotterized evolution lasts nearly twice as much time in absolute units, since the averaged J0 is roughly half as large, it nonetheless shows a substantial reduction in decoherence and improvement in fidelity to the desired Hamiltonian. An initial state with one spin flip is chosen for this comparison, as it makes the effect of decoherence due to phonons more pronounced compared with a state near zero net magnetization.

### Extended Data Fig. 3 Histograms of r.

Probability density distributions of r, the ratio of adjacent energy level spacings, for the experimental Hamiltonian (equation (1) of the main text) at various values of g/J0 and N = 15. Numerics are compared with the distribution expected for either a Poisson level distribution (blue lines in a and d) or a Wigner–Dyson distribution (red lines in b, c). The level statistics in the absence of a field gradient are near the Poissonian limit, which may reflect the proximity to an integrable limit for the low-energy sector68. A small gradient results in statistics near the Wigner–Dyson limit, followed by an approach to Poisson statistics as the gradient is increased.

### Extended Data Fig. 4 Dependence of ⟨r⟩ on α and g/J0.

Dependence of r on α and g/J0 (N = 13, Bz0/J0 = 5), for the power-law Hamiltonian (equation (30)). In the experiments presented in the main text α ≈ 1.3.

### Extended Data Fig. 5 Dependence of ⟨r⟩ on system size.

Level statistics for N = {9, 11, 13, 15} (light to dark), for α = 1.3 and Bz0/J0 = 5 and for the power-law Hamiltonian (equation (30)).

### Extended Data Fig. 6 Dependence of $$\bar{{\boldsymbol{ {\mathcal I} }}}$$ on system size and time.

a, Numerics showing $$\bar{ {\mathcal I} }$$ for the Néel state with N = {9, 15, 25} (light to dark). As the system increases from N = 9 to N = 25, the largest change is in a sharpening feature near g/J0 = 1. These numerics do not include experimental noise. b, Experimental data for N = 15 and N = 25, reproduced from Fig. 2, shows a similar dip for the larger size. c, Expanded view of numerics from a. Especially for gradient values above g/J0 = 1, the imbalance shows little finite-size dependence. d, Numerical comparison of $$\bar{ {\mathcal I} }$$ (N = 15) for the experimental time and for an extended time of 100tJ0 (dashed). While at small gradients the finite-time effects on the imbalance are substantial, including the dip feature in the left plots, a steady state is largely achieved in the experimental window for gradients g/J0 > 1. For all numerics shown, Bz0/J0 = 4.4(1 + 3g/(5J0)) (the experimental scaling resulting from equation (13) with Δt1 varied) and α = 1.3.

### Extended Data Fig. 7 Long-term stability of Stark MBL.

a, b, Numerical study of the long-time dynamics of the initial states realized in Fig. 2, using exact diagonalization. For this finite-size realization, in a strong gradient (g/J0 = 2, solid lines), the imbalance and bipartite entanglement entropy show some slow dynamics but apparently never approach the thermal value, in contrast with a weak gradient (g/J0 = 0.25, dashed line). c, Numerical study of the finite-size and initial-state dependence of Stark MBL imbalance dynamics. States with one-block (Néel) and two-block domain walls are shown for g/J0 = 2 and N = 12, N = 16, and N = 20 (light to dark solid lines, N = 20 for the two-block state only). The two-block initial state shows faster decay and greater finite-size effects, as is expected from the effective Hamiltonian in a large tilt (equation (26)). With a stronger gradient (dashed line, g/J0 = 5 and N = 12), this instability can be arbitrarily postponed. To show the long-term trend clearly, a moving average with a window of 5J0 has been applied to these numerics. d, Experimental data for the one and two-block domains. Consistent with numerics, state-dependent instability is manifested as a slow differential increase in the decay of the two-block state compared to the Néel state. These data were taken consecutively to ensure identical experimental parameters and decoherence rates. Each point is an average over 200 experimental repetitions, with error bars smaller than the symbol size. e, Numerical studies of stability in a quadratic field (N = 16, γ = 2) do not show this state-dependent instability over the same timescale. To show the long-term trend clearly, a moving average with a window of 5J0 has been applied to these numerics. f, Cartoon of the setup for numerics in e (shown with N = 8 for clarity). The quadratic potential is chosen to have a minimum shifted away from the system centre by one-quarter site to avoid a fine-tuned reflection symmetry. For all numerics shown, Bz0/J0 = 4.5 and α = 1.3.

### Extended Data Fig. 8 QFI.

Normalized QFI for a Néel state (N = 15) with g/J0 = 0.24 (white) and g/J0 = 2.4 (blue), corresponding to the lowest- and highest-gradient data in Fig. 2d. Points are experimental observations, averaged over 200 repetitions, with lines as guides to the eye. A value greater than one (dashed line) is an entanglement witness. After the initial fast dynamics up to tJ0 ≈ 1, the QFI is consistent with saturation for the small gradient, and with slow entanglement growth for the large gradient, with behaviour very similar to that previously observed in disordered MBL30.

### Extended Data Fig. 9 Additional DEER data.

DEER Difference signal for R = {1, 2, 3} (light to dark), compared with the imbalance $${\mathcal I} (t)$$ for the same parameters. Data are offset for clarity but otherwise share the same axes. $${\mathcal I}$$ is taken from the same dataset as the R = 1 spin-echo data, with the probe spin excluded from the imbalance calculation. After tJ0 ≈ 2, the imbalance is essentially constant at the low but finite steady-state value corresponding to this gradient strength. However, correlation dynamics are still progressing—in particular, correlations as measured by the difference signal only begin to develop for R = 2 after this point. This is similar to the disordered MBL state, in which slow entanglement dynamics continue after the locally conserved populations have reached a steady state10,11,44. Points are averaged over 2,000 repetitions, with error bars representing statistical uncertainty of the mean (1σ s.e.m.).

### Extended Data Fig. 10 Critical slope in quadratic field.

As the quadratic curvature is varied, the division between thermalizing and nonthermal regions is largely consistent with a critical slope near g/J0 = 0.5. However, the strongest curvature of γ = 3.6 deviates from this rule. For the lowest two values of γ the system was completely delocalized, and thus only the lower bound is meaningful. Points are averaged over 200 experimental repetitions. Error bars (aside from the first two points) denote a variation of ±1 spin location.

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Morong, W., Liu, F., Becker, P. et al. Observation of Stark many-body localization without disorder. Nature 599, 393–398 (2021). https://doi.org/10.1038/s41586-021-03988-0

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